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Properties

Label 2-722-19.4-c1-0-9
Degree $2$
Conductor $722$
Sign $0.998 + 0.0463i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.173 − 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)6-s + (0.5 − 0.866i)7-s + (−0.500 − 0.866i)8-s + (1.87 − 0.684i)9-s + (3 + 5.19i)11-s + (0.5 − 0.866i)12-s + (−0.868 + 4.92i)13-s + (−0.766 + 0.642i)14-s + (0.173 + 0.984i)16-s + (−2.81 − 1.02i)17-s − 2·18-s + (−0.939 − 0.342i)21-s + (−1.04 − 5.90i)22-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.100 − 0.568i)3-s + (0.383 + 0.321i)4-s + (−0.0708 + 0.402i)6-s + (0.188 − 0.327i)7-s + (−0.176 − 0.306i)8-s + (0.626 − 0.228i)9-s + (0.904 + 1.56i)11-s + (0.144 − 0.249i)12-s + (−0.240 + 1.36i)13-s + (−0.204 + 0.171i)14-s + (0.0434 + 0.246i)16-s + (−0.683 − 0.248i)17-s − 0.471·18-s + (−0.205 − 0.0746i)21-s + (−0.222 − 1.25i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.998 + 0.0463i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.998 + 0.0463i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16692 - 0.0270395i\)
\(L(\frac12)\) \(\approx\) \(1.16692 - 0.0270395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (0.173 + 0.984i)T + (-2.81 + 1.02i)T^{2} \)
5 \( 1 + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.868 - 4.92i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.81 + 1.02i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-2.29 - 1.92i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-8.45 + 3.07i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-6.12 + 5.14i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-2.29 - 1.92i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-8.45 - 3.07i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.66 + 6.42i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-4.69 + 1.71i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.59 + 3.85i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.21 + 6.89i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.73 - 9.84i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3 + 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.08 + 11.8i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (9.39 + 3.42i)T + (74.3 + 62.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.26353894478859634216136230418, −9.445648598735570629234962644427, −8.960432398691497268966266893361, −7.55161629199631025604623738969, −7.02874368161638985869440891252, −6.49583598994266168763927341432, −4.72018300970412051946087284715, −3.95902906327944848217397195939, −2.16816774011449608256489919894, −1.32619934332249299742012475508, 0.903095117112165033300385754387, 2.66486529464486256595206532921, 3.92285761863944884069613880240, 5.11152448676923494957027906561, 6.02855926015529490234122682378, 6.89940968447630694476901979949, 8.136478016147562150199423508202, 8.622544650381514622257568371929, 9.531696369412504975227840934105, 10.47838846157201727685343996300

Graph of the $Z$-function along the critical line